# Reserved recheck keys · staff only

Keep these keys separate from the four unused first sources and their change conditions. Prior answer access ends a fresh claim. Use the descriptions and support record in [KEYS.md](KEYS.md).

## E1 · Pattern tokens

Nine parts give 7 per part: 28 rings, 35 bars; fraction 4/9. A $0.40/ring, 4 packs 28, $11.20, zero spare. B $0.35/ring, 7 packs 28, $9.80, zero spare. Initially B fits $10 and A does not. Later B $11.80 exceeds $11.50; A $11.20 fits. Revise to A; counts and pack quantities are unchanged. Use the E criteria.


## E2 · Mark cards

Seven parts give 8 each: 24 dot, 32 dash; fraction 3/7. A $0.30/card, 5 packs 25, $7.50, one spare. B $0.29/card, 3 packs 24, $6.96, zero spare. Initially only B fits $7.20. Later B $7.76 exceeds $7.60; A $7.50 fits with $0.10 spare. Revise to A. Use the E criteria, paying attention to the whole-pack condition.


## F1 · Joined paper lengths

Five groups give 5 short, 15 long; short fraction 1/4. Length 5 × 2/5 + 15 × 3/5 = 2 + 9 = 11 m. Both need 11 whole metre units, no spare. A $17.60; B $15.40. Initially B fits $15.50, A does not. Later B $18.40 exceeds $18, A $17.60 fits. Revise to A. Use the F criteria.


## F2 · A paper track diagram

Six groups give 18 short, 6 long; fraction 3/4. Length 18 × 1/3 + 6 × 3/4 = 6 + 9/2 = 21/2 m = 10½ m. Buy 11 whole metre units, leaving 1/2 m. A $12.10; B $11. Initially only B fits $11.20. Later B costs $12.50 and A costs $12.10. Under the new $11.90 budget, neither fits: A exceeds it by $0.20 and B by $0.60. Keep the unchanged need and purchase quantity, then explain that no sufficient order is affordable. Use the F criteria, with particular attention to the discrete purchase versus continuous needed length.